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| Author | : Paolo Mancosu |
| Publisher | : Oxford University Press |
| Total Pages | : 431 |
| Release | : 2021 |
| Genre | : Philosophy |
| ISBN | : 0192895931 |
Download An Introduction to Proof Theory Book in PDF, ePub and Kindle
An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.
| Author | : Gaisi Takeuti |
| Publisher | : Courier Corporation |
| Total Pages | : 514 |
| Release | : 2013-01-01 |
| Genre | : Mathematics |
| ISBN | : 0486490734 |
Download Proof Theory Book in PDF, ePub and Kindle
Focusing on Gentzen-type proof theory, this volume presents a detailed overview of creative works by author Gaisi Takeuti and other twentieth-century logicians. The text explores applications of proof theory to logic as well as other areas of mathematics. Suitable for advanced undergraduates and graduate students of mathematics, this long-out-of-print monograph forms a cornerstone for any library in mathematical logic and related topics. The three-part treatment begins with an exploration of first order systems, including a treatment of predicate calculus involving Gentzen's cut-elimination theorem and the theory of natural numbers in terms of Gödel's incompleteness theorem and Gentzen's consistency proof. The second part, which considers second order and finite order systems, covers simple type theory and infinitary logic. The final chapters address consistency problems with an examination of consistency proofs and their applications.
| Author | : Ulrich Kohlenbach |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 539 |
| Release | : 2008-05-23 |
| Genre | : Mathematics |
| ISBN | : 3540775331 |
Download Applied Proof Theory: Proof Interpretations and their Use in Mathematics Book in PDF, ePub and Kindle
This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.
| Author | : Wolfram Pohlers |
| Publisher | : Springer |
| Total Pages | : 220 |
| Release | : 2009-06-10 |
| Genre | : Mathematics |
| ISBN | : 3540468250 |
Download Proof Theory Book in PDF, ePub and Kindle
Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.
| Author | : S.R. Buss |
| Publisher | : Elsevier |
| Total Pages | : 823 |
| Release | : 1998-07-09 |
| Genre | : Mathematics |
| ISBN | : 0080533183 |
Download Handbook of Proof Theory Book in PDF, ePub and Kindle
This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.
| Author | : K. Schütte |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 309 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642664733 |
Download Proof Theory Book in PDF, ePub and Kindle
This book was originally intended to be the second edition of the book "Beweis theorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten. As well as classical predicate logic we also treat intuitionistic predicate logic. The sentential calculus properties of classical formal and semiformal systems are treated using positive and negative parts of formulas as in the book "Beweistheorie". In a similar way we use right and left parts of formulas for intuitionistic predicate logic. We introduce the theory of functionals of finite types in order to present the Gi:idel interpretation of pure number theory. Instead of ramified type theory, type-free logic and the associated formalization of parts of analysis which we treated in the book "Beweistheorie", we have developed simple classical type theory and predicative analysis in a systematic way. Finally we have given consistency proofs for systems of lI~-analysis following the work of G. Takeuti. In order to do this we have introduced a constni'ctive system of notation for ordinals which goes far beyond the notation system in "Beweistheorie."
| Author | : Wolfram Pohlers |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 380 |
| Release | : 2008-10-01 |
| Genre | : Mathematics |
| ISBN | : 354069319X |
Download Proof Theory Book in PDF, ePub and Kindle
The kernel of this book consists of a series of lectures on in?nitary proof theory which I gave during my time at the Westfalische ̈ Wilhelms–Universitat ̈ in Munster ̈ . It was planned as a successor of Springer Lecture Notes in Mathematics 1407. H- ever, when preparing it, I decided to also include material which has not been treated in SLN 1407. Since the appearance of SLN 1407 many innovations in the area of - dinal analysis have taken place. Just to mention those of them which are addressed in this book: Buchholz simpli?ed local predicativity by the invention of operator controlled derivations (cf. Chapter 9, Chapter 11); Weiermann detected applications of methods of impredicative proof theory to the characterization of the provable recursive functions of predicative theories (cf. Chapter 10); Beckmann improved Gentzen’s boundedness theorem (which appears as Stage Theorem (Theorem 6. 6. 1) in this book) to Theorem 6. 6. 9, a theorem which is very satisfying in itself - though its real importance lies in the ordinal analysis of systems, weaker than those treated here. Besides these innovations I also decided to include the analysis of the theory (? –REF) as an example of a subtheory of set theory whose ordinal analysis only 2 0 requires a ?rst step into impredicativity. The ordinal analysis of(? –FXP) of non- 0 1 0 monotone? –de?nable inductive de?nitions in Chapter 13 is an application of the 1 analysis of(? –REF).
| Author | : Sara Negri |
| Publisher | : Cambridge University Press |
| Total Pages | : 279 |
| Release | : 2008-07-10 |
| Genre | : Mathematics |
| ISBN | : 9780521068420 |
Download Structural Proof Theory Book in PDF, ePub and Kindle
A concise introduction to structural proof theory, a branch of logic studying the general structure of logical and mathematical proofs.
| Author | : George Metcalfe |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 279 |
| Release | : 2008-11-27 |
| Genre | : Mathematics |
| ISBN | : 1402094094 |
Download Proof Theory for Fuzzy Logics Book in PDF, ePub and Kindle
Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.
| Author | : A. S. Troelstra |
| Publisher | : Cambridge University Press |
| Total Pages | : 436 |
| Release | : 2000-07-27 |
| Genre | : Computers |
| ISBN | : 9780521779111 |
Download Basic Proof Theory Book in PDF, ePub and Kindle
This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.
